An integrable discretization of KdV at large times

An 'exact discretization' of the Schrodinger operator is considered and its direct and inverse scattering problems are solved. It is shown that a diff erential-difference nonlinear evolution equation depending on two arbitrary constants can be solved by using this spectral transform and that for a sp ecial choice of the constants it can be considered an integrable discretiza tion of the KdV equation at large times. An integrable difference-differenc e equation is also obtained.